The coefficients \(r^{(\ell ),i,c}\)  are rational functions of \(s_{ij}\)  and linear functions of \({\text{tr}_5}\) . By \(t_e\)  we denote the elements of the transcendental basis. The parity-odd variable \({\text{tr}_5}\)  has two distinct origins: the helicity projection and the master integrals of ref. [1]}. Internally, we keep these two separate. We relabel the \({\text{tr}_5}\)  originating from the masters as \(\varepsilon _5\)  and include it as part of the basis \(t_e\) . Since \(\varepsilon _5\)  only appears together with parity-odd combinations of pentagon functions, absorbing \(\varepsilon _5\)  in the transcendental basis effectively renders all elements \(t_e\)  parity-even.
